By Abraham A. Ungar
Facts that Einstein's addition is regulated through the Thomas precession has come to gentle, turning the infamous Thomas precession, formerly thought of the gruesome duckling of exact relativity concept, into the gorgeous swan of gyrogroup and gyrovector area thought, the place it's been prolonged through abstraction into an automorphism generator, referred to as the Thomas gyration. The Thomas gyration, in flip, permits the creation of vectors into hyperbolic geometry, the place they're referred to as gyrovectors, in one of these manner that Einstein's pace additions seems to be a gyrovector addition. Einstein's addition hence turns into a gyrocommutative, gyroassociative gyrogroup operation within the related method that standard vector addition is a commutative, associative team operation. a few gyrogroups of gyrovectors admit scalar multiplication, giving upward thrust to gyrovector areas within the similar means that a few teams of vectors that admit scalar multiplication supply upward thrust to vector areas. additionally, gyrovector areas shape the environment for hyperbolic geometry within the comparable method that vector areas shape the environment for Euclidean geometry. particularly, the gyrovector house with gyrovector addition given through Einstein's (M?bius') addition types the atmosphere for the Beltrami (Poincar?) ball version of hyperbolic geometry.
The gyrogroup-theoretic concepts built during this e-book to be used in relativity physics and in hyperbolic geometry let one to resolve previous and new very important difficulties in relativity physics. A working example is Einstein's 1905 view of the Lorentz size contraction, which used to be contradicted in 1959 through Penrose, Terrell and others. the appliance of gyrogroup-theoretic ideas in actual fact tilt the stability in prefer of Einstein.
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Extra info for Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces
Observers at rest relative to S (relative to S¢) agree with observers ) that their space coordinates are parallel at rest relative to S¢(relative to S¢¢ (by realizing that their frames are linked by a Lorentz transformation without rotation). Yet, observers at rest relative to S agree with observers at rest relative to S¢¢that their space coordinates are in relative rotation. This relative rotation generated by u, v ˛ which does not exist in classical mechanics, is the relativistic rotation known as the Thomas precession, denoted gyr[u, v].
9 MATRIX REPRESENTATION OF THE THOMAS PRECESSION To allow matrix representations of linear maps, we assume in this section that the abstract real inner product space is realized by a Euclidean n-space, [Ung88a][Ung89b][Ung90b][Ung91d]. Vectors x in are represented by their Cartesian components relative to a given basis of which we write as column matrices so that their transposes, x t = (x 1 , x 2 , . . , x n ), are written as Thomas Precession: The Missing Link 25 we define the square n × n matrix row matrices.
Kronig saw the implications of W. Pauli’s proposition that the total angular momentum of an electron is larger than its orbital momentum. To give this momentum a dynamic interpretation, Kronig supposed the electron to spin about its own axis. However, when he 1 2 GYROGROUPS AND GYROVECTOR SPACES calculated a new effect in the fine structure of the hydrogen atom, he obtained twice the amount required by the observations: The observed fine-structure intervals were found to have one half the theoretically expected lengths.